A comment on the combination of the implicit function theorem and the Morse lemma

TL;DR

This paper extends the analytic implicit function theorem by integrating the function with respect to the dependent variable and provides a geometric interpretation using the Morse lemma, aiding in hierarchical minimization techniques.

Contribution

It introduces an extension of the implicit function theorem incorporating integration and offers a geometric perspective via the Morse lemma for hierarchical minimization.

Findings

Extended the implicit function theorem with integration.

Provided a geometric interpretation using the Morse lemma.

Applied the results to hierarchical minimization in nonlinear parameter identification.

Abstract

The analytic implicit function theorem is extended. The function f of the theorem is integrated with respect to the dependent variable of the implicit function. A geometrical interpretation is given for the sub-geometry of the integral function F by using the Morse lemma. The result is used in the analysis of the hierarchical technique related to the minimization inon-linear parameter identification.